Spaces of phylogenetic diversity indices: combinatorial and geometric properties

Abstract

Biodiversity is a concept most naturally quantified and measured across sets of species. Yet for some applications, such as prioritising species for conservation efforts, a species-by-species approach is desirable. Phylogenetic diversity indices are functions that apportion the total biodiversity value of a set of species across its constituent members. As such, they aim to measure each species' individual contribution to, and embodiment of, the diversity present in that set. However, no clear definition exists that encompasses the popular diversity indices in current use. This paper presents conditions that define diversity indices arising from the phylogenetic diversity (PD) measure on rooted phylogenetic trees. In this context, the diversity index 'score' given to a species represents a measure of its unique and shared evolutionary history as displayed in the underlying phylogenetic tree. Our definition generalises the diversity index notion beyond the popular Fair Proportion and Equal-Splits indices. These particular indices may now be seen as two points in a convex space of possible diversity indices, whose boundary conditions are determined by the underlying shape of each phylogenetic tree. We calculate the dimension of the convex space associated with each tree shape, and describe the extremal points.

The robustness of phylogenetic diversity indices to extinctions

“…Finally, phylogenetic diversity indices, as have been used until now and as were described in [16], have been based on the assumption that, while edge lengths are used to calculate particular score values, the method of calculation should be independent of the lengths themselves. This has not been adequately justified apart from on the grounds of mathematical simplicity and the fact that the two diversity indices used in practice (FP and ES) both have this property.…”

“…Let Γ i (v, e) = x∈Ti(v) γ(x, e), that is, the sum of all coefficients associated with both edge e and some leaf in the i-th pendant subtree below vertex v. Let e = (u, v) be an edge of T . Then the ratio Γ 1 (v, e) : • • • : Γ d (v, e) is called the ratio of allocations at v, where d is the out-degree of v. Moreover, diversity indices in this paper will be assumed to be consistent in the sense that the values Γ i (v, f ) lie in the same ratio as the ratio of allocations at v for every edge f in the path between the root vertex and v. Each diversity index may be described by its (consistent) ratios of allocations, with [16] showing both how these ratios may be converted into γ(x, e)-type coefficients and that all diversity indices may be expressed in a consistent form.…”

“…We set γ(x j , e + j ) = α, where α lies in the open interval (0, 1) because ϕ is not a boundary index. Let γ(x i , e + j ) = p. Since ϕ has a consistent form (Proposition 11 of [16]), we can write p = (1 − α)q for some q ∈ (0, 1). Increasing the length of e + j by, say, k units increases ϕ T (x i ) by pk but increases ϕ T (x i ) by qk, a different amount since α is nonzero.…”

“…Moreover, diversity indices in this paper will be assumed to be consistent in the sense that the values Γ i ( υ, f ) lie in the same ratio as the ratio of allocations at υ for every edge f in the path between the root vertex and υ . Each diversity index may be described by its (consistent) ratios of allocations, with [16] showing both how these ratios may be converted into γ ( x, e )-type coefficients and that all diversity indices may be expressed in a consistent form.…”

“…We call any diversity index that is not boundary an interior diversity index. Equivalently, a diversity index ϕ(x) = e∈E γ(x, e) (e) is interior if and only if γ(x, e) is strictly positive whenever x is descended from e. The names 'boundary' and 'interior' refer to the position of a diversity index in S(T, ), the compact convex space of diversity indices on the tree T under edge length assignment (see [16] for details).…”